There is no limit on rate of fire in the core rules
There is no limitation on the number of times a siege weapon can be fired in the core rules. In fact, those rules are so slim we can look at their entirety here (Siege Equipment section, page 255 DMG):
Siege weapons are designed to assail castles and other walled fortifications. They see much use in campaigns that feature war. Most siege weapons don't move around a battlefield on their own; they require creatures to move them, as well as to load, aim, and fire them.
That’s it. The rest of the section consists of a list of example weapons; here is the Ballista:
Armor Class: 15
Hit Points: 50
Damage Immunities: poison, psychic
A ballista is a massive crossbow that fires heavy bolts. Before it can be fired, it must be loaded and aimed. It takes one action to load the weapon, one action to aim it, and one action to fire it.
Bolt. Ranged Weapon Attack: +6 to hit, range 120/480 ft., one target. Hit: 16 (3dl0) piercing damage.
The actions needed to load, aim and fire the weapon are not its own actions. They are the operators' actions. While each operator may have only one action per turn, with more operators you can have more actions to operate the weapon. Indeed it is common practice to have a large enough crew so that they can fire the weapon each round by pooling their actions. There’s nothing in there that limits how many times it could be fired in a round with a large enough crew.
Reasonably, the crew needs to be next to the ballista to manipulate it, so there may be a ceiling by the movement of the crew and how many crew members could be near enough the weapon to move there, take their action, and move away again to make room for others. With a normal speed of 30 and medium size, this might be up to six sets of crew in combat, for a total of six attacks per round, or sixty per minute.1
As PlayPatrice points out in the comments, even the ten shots per minute you would get from one attack per round outpaces the rates of fire with more modern day artillery weapons. Historically ballistae could be fired about two times per minute, or once every 5 rounds, so the DM might - as the question suggests - want to apply common sense to impose a limit over what the mechanical rules would allow.
Restrictions for vessel-mounted siege weapons
There are limits to rate of fire per round for siege weapons that are part of a vessel, due to special vessel combat rules.
Ghosts of Saltmarsh has rules for siege weapons that are mounted on ships. While the weapons are the ones from the DMG, when they are part of a vessel, firing them becomes one of the vessels action's instead of using individual actions by crew members for individual weapons (and in fact, crew are not assigned to specific weapons). The vessel’s actions are limited by round. For example, here is the text from a Galley (p. 187):
On its turn, the galley can take 3 actions, choosing from the options below. It can take only 2 actions if it has fewer than forty crew and only 1 action if it has fewer than twenty. It can't take these actions if it has fewer than three crew.
Fire Ballistas. The galley can fire its ballistas (DMG, ch. 8). […]
And here is the Ballista's weapon feature of the Galley:
WEAPONS: BALLISTAS (4)
Armor Class 15
Hit Points 50 each
Ranged Weapon Attack: +6 to hit, range 120/480 ft., one target. Hit: 16 (3dl0) piercing damage.
As I read it, using the Fire Ballistas action will fire all four Ballistas, once. So, the four ballistas mounted on a Galley could each be at most fired three times per round, using all three of the Galley’s actions.
This still is not due to an inherent limitation on the siege weapon’s rate of fire per round, but due to a limitation on how many actions are available each round to fire it.
1 Maybe you can construct better rotation patterns with enough space around the weapon — I’ll skip this optimization problem. Often the weapons will be on battlements or cluttered ship decks with limited space to spare.